In the preceding Sections REF  and REF  all symbols we have studied are defined in some conic (in \(\xi \) ) neighborhood of \((X,\epsilon )=({\bar{X}},0)\)  or \(X={\bar{X}}\) . To make notations clear, in this Section, we denote such symbols defined microlocally by putting tildes such that \({\tilde{a}}(X,\epsilon )\)  or \({\tilde{b}}(X)\)  so on and we extend such symbol to \({\mathbb {R}}^d\times {\mathbb {R}}^d\)  following [1]} (also [2]}).
